Theorem 0nnq 10815

Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
0nnq	⊢ ¬ ∅ ∈ Q

Proof of Theorem 0nnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5648	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 10803	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3	2	ssrab3 4029	. . 3 ⊢ Q ⊆ (N × N)
4	3	sseli 3925	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
5	1, 4	mto 197	1 ⊢ ¬ ∅ ∈ Q

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2111  ∀wral 3047  ∅c0 4280   class class class wbr 5089   × cxp 5612  ‘cfv 6481  2^nd c2nd 7920  Ncnpi 10735   <_N clti 10738   ~_Q ceq 10742  Qcnq 10743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-opab 5152  df-xp 5620  df-nq 10803

This theorem is referenced by:  adderpq  10847  mulerpq  10848  addassnq  10849  mulassnq  10850  distrnq  10852  recmulnq  10855  recclnq  10857  ltanq  10862  ltmnq  10863  ltexnq  10866  nsmallnq  10868  ltbtwnnq  10869  ltrnq  10870  prlem934  10924  ltaddpr  10925  ltexprlem2  10928  ltexprlem3  10929  ltexprlem4  10930  ltexprlem6  10932  ltexprlem7  10933  prlem936  10938  reclem2pr  10939